Notation
\(r_{A}\) and \(r_{B}\) denote the returns of A and B, and \(\bar{r}_{A}\) and \(\bar{r}_{B}\) denote their means (expected returns).
\(w_{A}\) and \(w_{B}\) denote portfolio weights = allocations
(like 0.4 and 0.6).
Portfolio return is \(r_{p}=w_{A}r_{A}+w_{B}r_{B}\)
In general, \(r_{p}=\sum_{i=1}^n w_ir_i\) and weights sum to 1.
The Mean Portfolio Return is . . .
\[\bar{r}_p=w_{A}\bar{r}_A+w_{B}\bar{r}_B\]
In general, \(\bar{r}_p=\sum_{i=1}^n w_i\bar{r}_i\)
Portfolio Variance
Variance means squared deviation from mean, so
\[var(r_{p})=E[(r_{p}-\bar{r}_p)^2]\]
A line of algebra shows that the deviation from the mean \(r_{p}-\bar{r}_{p}\) is a weighted average of deviations from means:
\[r_{p}-\bar{r}_{p}=w_{A}(r_{A}-\bar{r}_A)+w_{B}(r_{B}-\bar{r}_B)\]
So, the squared deviation is
\[\small w^{2}_{A}(r_{A}-\bar{r}_{A})^2+w^{2}_{B}(r_{B}-\bar{r}_{B})^2+2w_{A}w_{B}(r_{A}-\bar{r}_{A})(r_{B}-\bar{r}_{B})\]
Taking the expectation, the first two terms are variances and the third is a covariance.
Portfolio Variance with Two Assets
\[\small var(r_{p})=w^2_{A}var(r_{A})+w^2_{B}var(r_{B})+2w_{A}w_{B}cov(r_{A},r_{B})\]
- \(var(r_{p})\) is smaller when the covariance is smaller.
- Covariance is correlation times product of standard deviations, so \(var(r_{p})\) is smaller when the covariance is smaller.
Portfolio Variance with n Assets
\[\small var(r_{p})=\sum_{i=1}^n w_{i}^2var(r_i)+2\sum_{i=1}^n \sum_{j=i+1}^n w_{i}w_{j}cov(r_{i},r_{j})\]
There are \(\frac{n(n-1)}{2}\) covariance terms.
A Simple Case
\(n\) assets, all variances are the same = \(\sigma^2\),
all correlations are the same = \(\rho\),
all weights = \(\frac{1}{n}\).
\[\small var(r_{p})=\sum_{i=1}^n (\frac{1}{n})^2\sigma^2+2\sum_{i=1}^n \sum_{j=i+1}^n (\frac{1}{n})^2\rho\sigma^2\]
\[\small var(r_{p})=\frac{1}{n}\sigma^2+\frac{n-1}{n}\rho\sigma^2 \rightarrow \rho\sigma^2\]
3C
2A
Portfolio Std Dev < Weighted Average of Asset Std Devs
\[\small (w_{A}\sigma_{A}+w_{B}\sigma_{B})^2=w_{A}^2\sigma_{A}^2+w_{B}^2\sigma_{B}^2+2w_{A}w_{B}\sigma_{A}\sigma_{B}\]
same as \(var_{p}\) except no correlation in the last term. So,
\[(w_{A}\sigma_{A}+w_{B}\sigma_{B})^2>var(r_{p})\]
Some Diversification Usually Lowers Risk
Compare holding only asset B to holding some mix of A and B.
Substitute \(w_{B}=(1-w_{A})\). Variance in general is
\[\small w_{A}^2\sigma_{A}^2+(1-w_{A})^2\sigma_{B}^2+2w_{A}(1-w_{A})\rho\sigma_{A}\sigma_{B}\]
Derivative with respect to \(w_{A}\) evaluated at \(w_{A}\)=0:
\[\small 2w_{A}\sigma^2_{A}-2(1-w_{A})\sigma^2_{B}+2(1-2w_{A})\rho\sigma_{A}\sigma_{B}=-2\sigma^2_{B}+2\rho\sigma_{A}\sigma_{B}\]
Risk is lowered by \(w_{A}\) \(\uparrow\) when \(\rho\sigma_{A}<\sigma_{B}\)
